Understand that between the time Anakin pledges his life to Darth Sidious, and when he resurfaces into the "public" eye as Darth Vader behind the mask and suit, very few people see him and live. The Jedi in the temple are wiped out. The Separatist leaders are eradicated. The only ones I can think of who have seen Anakin between his fall and his transformation and aren't shown to be dead by the end of Ep III are Obi-Wan, Palpatine's personal guard, and a few clone troopers. The latter two categories are expendable if necessary. The medical droids that worked on him can easily have their memories reset. Yoda knows what happened to Anakin, but only ever sees Anakin in a recorded security hologram exterminating the Jedi and younglings in the Temple.
Also, keep in mind that the people to whom the secret of Vader's past is most important to keep are the Jedi, not the Empire. Luke cannot know Vader was Anakin until he is ready to know this. Luke uses the Skywalker name growing up; on Tatooine at least, that's not incredibly dangerous, as Shmi Skywalker was known to be Cleeg Lars' wife, and a slave before that; who knows who Watto made her shack up with to pay or buy someone off? But, if he had heard, or grew up knowing, that Vader's true name is Anakin Skywalker, one of two things would happen; he would either disavow the Force altogether, knowing what it did to his father, or he would seek his father out before he was ready. Frankly speaking, being as naive as he was he might have started asking questions about the Skywalker name that would work their way back to the Emperor, putting Luke in serious danger long before he ever hooks up with the Rebels.
By contrast, it's merely somewhat convenient that Vader's backstory isn't more public in-universe; if the galaxy knew that one of the most feared figures at the Emperor's right hand was Anakin Skywalker, formerly one of the most powerful Jedi in the galaxy, it would only heighten the fear surrounding Vader. As it is, having people not know he was even human can be useful for the same end, so it's merely never mentioned.